Quas
Quasii-equ
equili
equ
ilib
briu
rium Theo
Theo
eorry of Small
Perturbations to Radiative-
Convective Equilibrium States
States
• See Quasi
Quasi-Equilibrium
Equilibrium Dynamics of the
Tropical Atmosphere paper on course
web site
• Free troposphere assumed to have moist
adiabatic lapse rate (s*
(s does not vary
with height
• Boundary layer quasi
quasi-equilibrium
equilibrium applies
1
Basis of statistical equilibrium phys
ysiics
cs
• Dates to Arakawa and Schubert (1974)
• Analogy to continuum hypothesis:
Perturbations must have space scales >>
pacing
g
intercloud sp
• TKE consumption by convection ~ CAPE
generation by
ylar ge scale
• Numerical models on the verge of
gclouds + lar ge-scale waves
simulating
• We further assume convective criticality
2
Implications of the moist adiabatic lapse ra
ratte for th
the ssttruct
ructure of
of
trop
tropical disturbances
disturbances
• Approximate moist adiabatic condition as
that of constant saturation entropy:
Lv q *
p




T
 Rd ln 

s*  c p ln 


T
 T0 
 p0 
• Assume
ssu e hyyd
drostatic
ostat c pe
erturbations:
tu bat o s
 '
  '
p
3
• Maxwell’s
M
ll’ rellation:
ti
 T 
  
'
 s *'
 s *'  
 s *  p
 p  s*
• Integrate:
 '  b '( x, y, t )  T ((xx, y, t )  T  s *'
Only barotropic and first baroclinic mode survive
4
This implies, through the linearized momentum equations,
e.g.
u


 fv
fv
t
x
that the horizontal velocities may be partitioned similarly:
u  ub ( x, y, t )  T ( x, y, t )  T  u *( x, y, t );
v  vb ( x, y, t )  T ( x, y, t )  T  v *( x, y, t ).
5
6
Implications for vertical structure of vertical
i l velocity
l i
 u v 

   
pp
y 
 x y
Integrate:
 ub vb 
   p0  p  


 x y 
 p  p T  
0
7
p0
p

 u * v * 
Tdp ' 

 .
y 
 x
At tropopause:
 ub vb 
t   p0  pt  


 x y 
This implies that if a rigid lid is imposed at the
tropopause, the divergence of the barotropic
velocities must vanish and the barotropic
components therefore satisfy the barotropic
vorticity equation:
b
 Vb ,
tt
ˆ   V  2 sin 
  k
b
b
8
9
Feedback of Air Motion on (virtual) Temperature
• Convection cannot change vertically
integrated enthalpy, k  c pT  Lv q
• Then neglecting surface fluxes
fluxes, radiation
radiation,
and horizontal advection,

h
k dp     dp,

t
p
• Neelin and Held (1987): This function is
gative for up
pward motion
neg
10
• Upward motion is associated with column
column
moistening:
T

q
c p t dp  t k dp   Lv t dp
Ascent leads to cooling
• Yano and Emanuel, 1991:
N
2
eff
 1   p  N
11
2
Prediction: Inviscid, small
amplitude perturbations under rigid
lid: Shallow water solutions with
reduced equivalent depth
12
Quasi-Linear  Plane System , N l i B
Neglecting
Barotropic
i M
Mode
d u
u
s *
 Ts  T 
  yv  ru
t
x
v
s *
 Ts  T 
  yu  rv
t
yy
sd
s *  d  


 p M  w 

 Qrad 

t
m 
z

sb
h
 Ck | V |  s0 * ssb    M  w   sb  sm 
t
13
u v w
  0
x y H
Quasi-Eq
quilibrium Assump
ption:
sb s *

t
t
Gives closure for convective mass flux,
M
M
System closed except for specification of

Qrad , s0 *, sm ,  p
14
Additional Approximations:
• Boundary Layer QE (Raymond,
(Raymond 1995):
1995):
Neglect h sb , gives simpler expression
t
for M
s0 * ssb
M  w  Ck | V |
sb  sm
• Weak
W kT
Temperature
t
Approxi
A
imation
ti
(Sobel and Bretherton, 2000): Neglect s * t
(Over-d
(O
determi
t
ined
d system,
t
ignore
momentum equation for irrotational flow)
15
Important Feedbacks:
• Wind
Wind-Induced
-Induced Surface Heat Exchange
(WISHE) Coupling of surface enthalpy flux
to wind perturbations (Neelin et al
al. 1987
1987,
Emanuel, 1987)
• Moisture-Convection
Moisture Convection Feedback:
Dependence of sm on M and/or  p on s * sm
• Cloud-Radiation
Cl d R di ti Feedback
F db k:
Dependence of Q rad on M or s * sm
16
• Ocean
Ocean-Atmosphere
-Atmosphere Feedback (e.g.
(e g
ENSO): Feedback between perturbation
surface wind and ocean surface
surface
temperature, as represented by s0 *
17
Simple Example:
•
•
•
•
Ignore perturbations of Q rad
Ignore fluctuations of  p
Make boundary layer QE approximation
Fully linearize surface fluxes:
| V |  U u*
2
Uu '
|V|'
|V|
18
2
Introduce scalings:
First define a merdional scale,
Ly :
d
sd 1   p
Ly 
Ts  T  H

m
z  2
4
Then let
xa x
y  Ly y
a
t
t
2
 Ly
aCk | V |
u
u
H
v
L y Ck | V |
H
s* 
v
19
aCk | V |  Ly
H Ts  T 
2
s *
Separate scalings for ocean temperature and lower
tropospheric entropy:
so * 
sm 
1  p
p
s
b

 s
 s * s *
1  p sb  sm
1
p

 sm s0
2
m
0
20
Nondimensional parameters:




1  p aCk U s0 * s *

 p H | V | s * ssm
ra

 Ly 2
2

( Rayyleigh
g friction
f
)
 s * ss *
H T  T   s * s 
11  p aCk | V |  Ly
p
 a
 
L
 y
(WISHE)
2
0
2
s



m
2
(zonal geostropy )
21
(surface damping)
Nondimensional Equations:
u s
  yv  u
t x
 s

v
    yu   v 
t
 y

s u v

   u  s0  sm   s
t x y
22
Steady System with   sm  0 :
s
s
2
2
  y   y s   y s0
x
y
Similar to Gill Model, but forcing is directly in
terms of SST (s0), not latent heating
For SST of the form
s0  RE G(
G ( y )e 
ikx
there are solutions of the form
ikkx

s  RE  J ( y )e  ,
23
where
J ( y)  y
ik

 y2
e
2

y
0
Gu
Example:
Ge
24
by 2
1 ik

e
2

u

2
du
  0,
k  2, b  1.5,   1.5
25
  1,
1,
k  2,
2, b  1.5
1.5,,   1.5
26
Basic linear wave dynamics on the equatorial
i l β plane
l
Omit damping and WISHE terms from linear
nondimensional equations:
u s
  yv
t x
 s

v
    yu 
t
 y

ss u
u v
v


t x y
27
Fully equivalent to the shallow water equatiions on a β pllane
Eliminate s and u in favor of v:
  v  v
v
v
v
2 
0
 2  2   2   y v  
t  t
x
y
x

2
Let
2
v  V ( y )e
2
ikx it
2
2

k
 k
dV
2
 2 
  y V  0
d
dy

 

2
28
Boundary conditions: V well behaved at
y  
Solution in terms of discrete parabolic cylinder functions Dn :
v  Dn ( y ),
whhere Dn  e
2
2 y2
1, 2 y, 4 y 2  22,

provided ω satisfies the dispersion relation
 k
k
  2n 1


2
2
29
There is, in addition, another mode satisfying v=0
everywhere.
h
F
From fifirstt and
d third
thi d linear
li
equations:
ti
 2u  2u
 2  0.
2
t
x
Satisfied by u  F  x  t  .
Eastward-propagating, nondispersive
equatorially trapped Kelvin wave
Note that this happens to satisfy derived dispersion relation when n= -1.
30
There are three roots of the general dispersion relation:
2  k2 k
n  0:
  1  0


   k 1

Factor : 
    k   0

 
  kk roott nott allowed
ll
d  does nott satisfy
ti f BCs
BC 


1
  k  k 2  4
2

Mixed Rossby-Gravity Waves (MRG)
31
For n ≥ 1, two well defined limits:
1. |  || k |
|:

k
k
2n  1  k
2


2. |  ||| k |:
|
  k   (2n  1)
2
2
32
Pl
t
Planetary
Rossby waves
Inertia-gravit
Inertia
gra ity
waves
33
Kelvin wave
wave
34
Mixed R
Rossby
ossby--Gravity
ossby
Gravity
35
Rossby
Rossby
36
Inertia--gravity
Inertia
gravity
37
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http://ocw.mit.edu
12.811 Tropical Meteorology
Spring 2011
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